ACTAUNIVERSITATIS

UPSALIENSISUPPSALA

2014

Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Social Sciences 97

On Non Parametric Regressionand Panel Unit Root Testing

XIJIA LIU

ISSN 1652-9030ISBN 978-91-554-8938-0urn:nbn:se:uu:diva-222242

Dissertation presented at Uppsala University to be publicly examined in Ekonomikum,Uppsala, Monday, 26 May 2014 at 10:15 for the degree of Doctor of Philosophy. Theexamination will be conducted in English. Faculty examiner: Niklas Ahlgren (Hanken Schoolof Economics).

AbstractLiu, X. 2014. On Non Parametric Regression and Panel Unit Root Testing. DigitalComprehensive Summaries of Uppsala Dissertations from the Faculty of Social Sciences 97.30 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-554-8938-0.

In this thesis, two different issues in econometrics are studied, the estimation of regressioncoefficients and the non-stationartiy analysis in a panel setting.

Regarding the first topic, we study a set of measure of location-based estimators (MLBEs) forthe slope parameter in a linear regression model with a single stochastic regressor. The median-unbiased MLBEs are interesting as they can be robust to heavy-tailed and, hence, preferableto the ordinary least squares estimator (LSE) in such situations. Two cases, symmetric stableregression and contaminated normal regression, are considered as we investigate the statisticalproperties of the MLBEs. In addition, we illustrate how our results can be extended to includecertain heteroscedastic regressions.

There are three papers concerning the second part. In the first paper, we propose a novel wayto test the unit roots in the panel setting. The new tests are based on the observation that thetrajectory of the cross sectional sample variance behaves differently for stationary than for non-stationary processes. Three different test statistics are proposed. The limiting distributions arederived and the small sample properties are studied by simulations. In the remaining papers, wefocus on the studies of the block bootstrap panel unit root tests proposed by Palm, Smeekes andUrbain (2011) which aims at dealing with a rather general cross-sectional dependency structure.One paper studies the robustness of PSU tests by a comparison with two representative testsfrom the second generation panel unit root tests. In another paper, we generalized the blockbootstrap panel unit root tests in the sense of considering the deterministic terms in the model.Two different methods to deal with the deterministic terms are proposed and the asymptoticvalidity of bootstrap tests under the main null hypothesis is theoretically checked. The smallsample properties are studied by simulations.

Xijia Liu, Department of Statistics, Uppsala University, SE-75120 Uppsala, Sweden.

© Xijia Liu 2014

ISSN 1652-9030ISBN 978-91-554-8938-0urn:nbn:se:uu:diva-222242 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-222242)

To my parents, wife and son

List of papers

This thesis is based on the following papers, which are referred to in the textby their Roman numerals.

I Liu, X., and Preve, D. (2012) Measure of location-based estimators insimple linear regression. Submitted.

II Liu,X. (2012) Panel unit root tests based on sample variance.Submitted.

III Liu, X., and Wei, J. (2014) On the robustness of the block bootstrappanel unit root test.

IV Liu, X., and Wei, J. (2014) Block bootstrap panel unit root tests withdeterministic terms.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.1 The estimation of regression coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Heavy tailed distribution and stable distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Unit root tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4 Panel unit root tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Panel unit root tests with cross-sectional dependence . . . . . . . . . . . . . . . . . 131.6 Bootstrap method and bootstrap test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.7 Robust block bootstrap panel unit root tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Summary of the papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.1 Paper I: Measure of location based estimators in simple linear

regression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2 Paper II: Panel unit root tests based on the sample variance. . . . . . . 192.3 Paper III: On the robustness of the block panel unit root test . . . . . 222.4 Paper IV: Block bootstrap panel unit root tests with

deterministic terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Further research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1. Introduction

1.1 The estimation of regression coefficientsLinear models are widely applied in statistics and econometrics. Suppose theobservations are generated by a simple linear regression model

yi = α +βxi +ui, (1.1)

then the question is how to estimate the regression coefficients? The first ref-ereed classical solution must be the least squares estimator (LSE) created byCarl Friedrich Gauss. By minimizing the sum of squares of residuals, the LSEfor β in (1.1) is

βLS = ∑ni=1(xi− x)(yi− y)

∑ni=1(xi− x)2 , (1.2)

There are other estimators of the slope coefficient, for example, the maximumlikelihood estimator which is equivalent to LSE under the normality assump-tion. The LSE has a nice property provided by the Gauss-Markov theorem. Ifthe explanatory variable is non-stochastic and the regression errors are uncor-related random variables with zero mean and common finite variance, βLS isthe most efficient among all linear unbiased estimators for β . In other words,the LSE has the smallest variance in the class of linear unbiased estimatorsif certain relatively restrictive conditions are satisfied. However, those con-ditions are quite strong and unrealistic in some empirical works. In financialeconometrics, for example, researchers often work with the heavy tail distri-bution. (It will be discussed more in the next subsection). In this case, therewill be some large values of the error term and the assumption that the errorterms have finite variance may not be satisfied. Then the LSE is not the mostefficient estimator anymore. To solve this problem, some estimation methodsbased on non-parametric or distribution free techniques have been invented,for example the Theil-Sen estimator, see Sen (1968).

1.2 Heavy tailed distribution and stable distributionIn probability theory, a heavy tailed distribution is a distribution whose tailsare not exponentially bounded, i.e. the tails are heavier than the tails for theexponential distribution. A heavy tailed distribution provides a fundamentaltool in the study of rare events. In the case for which the extreme values occur

9

with relatively high probability, heavy tailed distributions can be applied tomodel those phenomena very well, for example in finance, economics, com-puter science and so on.

One class of the commonly used heavy tailed distributions is the family ofstable distributions. Stable distributions are widely applied in financial mod-eling because they generalize the normal (Gaussian) distribution and allowheavy tails and skewness. The reason for the term stable is that they retainthe shape (up to scale and shift) under addition: if X and Xin

i=1 are indepen-dent, identically distributed stable random variables, then for every n, ∑

ni=1 Xi

is equal to cnX + dn in distribution for some constants cn > 0 and dn. Thesimplest example of a stable distribution is the normal distribution, where thesum of independent identically normally distributed random variables is stillnormal with the same mean and variance by some normalization.

Generally, the distribution of a stable random variable is described by fourparameters, here denoted by a,b,c and d. The parameter a, the index of stabil-ity, is confined to the interval (0,2]. The skewness parameter b is confinedto [−1,1]. The scale parameter c > 0, and the location parameter d cantake on any real value. There exist a number of different parametrizationsfor symmetric stable distributions. In this thesis, we will use the S (a,b,c,d)parametrization in Definition 1.7 of Nolan (2012). This class may be definedby the characteristic function,

ϕ(t) = E (eitv) = e−ca|t|a+idt , (1.3)

where t is a real number. A random variable v is S (a,0,c,d) distributed ifits characteristic function is given by (1.3), while there is no general closedform expression for the density of a stable random variable. Beside the nor-mal distribution, there are only two cases for which closed form expressionsfor the density of a stable distributed random variable exist. These are theCauchy S (1,0,c,d) and the Levy S (1/2,1,c,d). For our study, the mostuseful properties are summarized as a Lemma in Paper I, and the reader isreferred to Nolan (2012), Nolan (2013) and Zolotarev (1986) for details.

1.3 Unit root testsIn time series econometrics, the non-stationary processes can be modeled bytime trend models or unit root models. It is important to distinguish these twodifferent non-stationary processes for many reasons both in terms of empiri-cal analysis and from the technical point of view. From the empirical pointof view, the unit root process has a special feature regarding the persistenceof innovations which may have important implications for the formulation ofeconomic models. More specifically, macroeconomists are interested in mak-ing an accurate judgment to see whether economic recessions have permanentconsequences for the level of future GNP, or if only the temporary downturns

10

with the lost output eventually made up during the recovery, Hamilton (1994).From the technical point of view, while, the presence of unit roots can resultin spurious inference in regression analysis cf Granger and Newbold (1974).Simply speaking, if you repeat doing regression on two independent randomwalks 100 times, then, at the 5 percent level, you will probably get signifi-cant regression around 70 times. The most well known test for unit roots isthe Dickey-Fuller test (DF test), see Dickey and Fuller (1979). Suppose theobservations are generated by the AR(1) process

yt = ρyt−1 +ut , (1.4)

where utiid∼ N(0,σ2), and y0 = 0. The null hypothesis H0 : ρ = 1 is tested

against H1 : ρ < 1. The test statistic is the LSE of ρ

ρ = ∑Tt=1 yt−1yt

∑Tt=1 y2

t−1, (1.5)

Under the null hypothesis, the LSE of ρ is super-consistent, i.e. ρ convergesto 1 at a rate 1/T . By proper normalizing, the asymptotic distribution of ρ isnot normal but a non-standard distribution which can be expressed in terms offunctional of Brownian motion.

T (ρ−1) L→12

W (1)2−1

∫ 1

0 [W (r)]2 dr(1.6)

where W (r) is Brownian motion process. The unit root hypothesis is testedbased on this limiting distribution which can be tabulated by Monte Carlosimulations. Dickey and Fuller (1979) and Said and Dickey (1984) consideredthe general serial correlation contained in the error terms and generalized itas the augmented Dickey-Fuller test (ADF test). Similarly, a semi-parametrictest procedure which also considered the serial correlation was proposed byPhillips and Perron (1988). Additionally it is worth mentioning the test pro-cedure proposed by Kwiatkowski, Phillips, Schmidt and Shin (1992), eventhough its null hypothesis is not unit root but stationarity. There are also someother unit root test procedures, for example the variance ratio test proposedby Lo and MacKinlay (1988). However, the most serious problem is thatthe power properties of all the existing tests are very poor especially whenthe sample size is small. Many researchers have endeavoured to improve thepower properties, and one interesting effort is panel unit root tests.

1.4 Panel unit root testsAfter the seminal work by Levin and Lin (1993), a number of authors havetried to improve the performance of unit root tests by adding the cross sectional

11

dimension, i.e. the panel unit root test (see Breitung and Pasaran (2008) for anoverall review). The first two most well known papers are Levin, Lin and Chu(2002) (LLC) and Im, Pesaran and Shin (2003) (IPS). They assume that timeseries yi0, ...,yiT on the cross section units i = 1,2, ...,N are generated by asimple AR(1) process for each i, which can be expressed as simple Dickey-Fuller regressions

∆yit =−ρiµi +ρiyi,t−1 + εit (1.7)where ∆yit = yit−yi,t−1. They further assume that the error terms εit are inde-pendent for all i and t. This crucial assumption can be seen as the main symbolof the first generation of panel unit root tests. The null hypothesis of interestis

H0 : ρ1 = ... = ρN = 0. (1.8)For the alternative hypothesis, LLC and IPS consider the following two hy-potheses respectively:

H1a : ρ1 = ... = ρN ≡ ρ and ρ < 0, (1.9)

H1b : ρ1 < 0, ...,ρN0 < 0, for some N0 ≤ N. (1.10)Under H1a, LLC assumed that the autoregressive parameter is identical for allcross section units. This is called the homogeneous alternative. Then the teststatistic pools the observations across the different cross section units as

τρ =∑

Ni=1 σ

−2i ∆y′iMτyi,−1√

∑Ni=1 σ

−2i

(y′i,−1Mτyi,−1

) (1.11)

where ∆yi =(∆yi1, ...,∆yiT )′, yi,−1 =(yi0, ...,yi,T−1)′, Mτ = IT−τT (τ ′T τT )−1

τ ′T ,τT is a T ×1 vector of ones,

σ2i =

∆y′iMi∆yi

T −2, (1.12)

where Mi = IT −Xi (X′iXi)−1 X′i, and Xi = (τT ,yi,−1). Under H1b, IPS as-

sumed that N0 of the N (0 < N0 ≤N) panel units are stationary with individualspecific autoregressive coefficients. This is referred to as the heterogeneous al-ternatives. For the construction of the test statistic, IPS suggests the mean ofthe individual specific t-statistics

τ =1N

N

∑i=1

τi (1.13)

where τi is the Dickey-Fuller t-statistic of cross section unit i. In additionto improving the performance of unit root tests, there also are some otherbenefits by adding panel units. For example, people can do a set of unit roottests simultaneously. Another advantage is that the asymptomatic normalitycan be achieved again in most cases.

12

1.5 Panel unit root tests with cross-sectional dependenceAlong with achieving the benefits of adding panel units, it also gives us sometroubles. For example, a proper limit theory has to take into account the re-lationship between the increasing number of time periods and cross sectionunits. Phillips and Moon (1999) subdivide the asymptotic behaviour into threekinds, sequential limit, diagonal limit and joint limit. They argued that thesequential limiting distribution is easy to achieve, but it does not necessarilyimply the joint limiting distribution. A simple method of deriving the jointlimiting distribution was provided. A more serious problem, however, is theassumption of independence of panel units. O’Connell (1998) showed that thegood size and power properties will be violated if the convenient but unrealis-tic assumption of cross sectional independence is excluded. To overcome thisproblem, the so called second generation panel unit root tests have emerged.Some studies considered introducing cross-sectional dependence by augment-ing the covariance matrix as an arbitrary positive definite matrix from a simplepositive definite diagonal matrix. Under this assumption, some panel unit roottests are proposed, for example, introducing the non-linear instrumental vari-able in Chang (2002), applying the sieve bootstrap method in Chang (2004), arobust t test in Breitung and Das (2005), and so on.

Another more popular way is to model the cross-sectional dependence bya factor structure. Assume that the observations can be modeled by Equation(1.7), but the assumption of the independence of cross section units is relaxedand instead the error term can be written as

εt = Γft +ξt (1.14)

where εt = (ε1t , ...,εNt)′, ft is an m× 1 vector of serially uncorrelated unob-

served common factors, and ξt = (ξ1t , ...,ξNt)′ is an N× 1 vector of serially

uncorrelated errors with mean zero and a positive definite covariance matrixΩξ , and Γ is an N×m matrix of factor loadings defined by Γ = (γ1, ...,γN)′.Without loss of generality the covariance matrix of ft is set to Im, and it isassumed that ft and ξt are independently distributed. Intuitively, the cross-sectional dependence is specified by the factor structure, and all the drawbacksdue to the cross-sectional dependence are gathered in the nuisance parameters,factor loadings. Then different methods are proposed to remove the nuisanceparameters. Moon and Perron (2004) applied a modified version of the princi-pal component approach to estimate the matrix of factor loadings and get thede-factored panel data by projecting the panel data into the space orthogonalto the factor loadings, then they performed a pooled DF test; Pesaran (2007)considers the single factor case and proxies the common factor by the cross-sectional averages. Another more influential approach is the so called PANICprocedure 1, proposed by Bai and Ng (2004). They consider a more generalsituation by allowing the non-stationarity to enter not only the idiosyncratic

1PANIC: Panel Analysis of Nonstationarity in Idiosyncratic and Common components

13

errors but also the common factors. Under this special framework, one couldexpect to overcome the difficulty that panel unit root tests may be severelybiased if the panel units are cross-cointegrated, see Banerjee, Marcellino andOsbat (2005). As argued in Bai and Ng (2010), Moon and Perron (2004) andPesaran (2007) all can be seen as a special case of the PANIC model under thehomogeneous setting. Two asymptotic independent tests for unit roots in dy-namic factors and uncorrelated errors are created. For the dynamic factors, ifonly one dynamic factor exists, the pooled DF test can be applied; If the num-ber of dynamic factors is larger than 1, they apply the common trend tests andJohansen’s cointegration methodology, see Johansen (1995), to the dynamicfactors.

1.6 Bootstrap method and bootstrap testIn order to introduce the main research subject of Paper III and IV, we willbriefly discuss the main ideas of an important tool in statistical inference,the bootstrap method. Almost all of the problems of statistical inferencecan ultimately be reduced to the understanding of the sampling distribution.Let Xin

i=1 be a random sample with common distribution F and let Tn =Tn(X1, ...,Xn) as a statistic of interest. In order to control the uncertainty anddraw a statistical inference, the sampling distribution of the statistic Tn is cru-cial. The main obstacle, however, is the unknown distribution function F .Even if the distribution function is known, it is still difficult to find the ex-act distribution for a given sample size because of the limitation of analyticalabilities. For the latter case, the situation is not too bad since one could findthe exact distribution using Monte Carlo simulations. More specifically, wecan repeatedly generate the realizations of a random sample Xin

i=1 from thecommon distribution function F , and then calculate the statistic Tn for eachsample and sequentially get the approximation of the distribution of statis-tic Tn. While, in the most tricky situation, if the distribution function is totallyunknown, then it is impossible to generate the random sample from F . In clas-sical statistical inference, the asymptotic approximation which only dependson some moment conditions is always the first alternative.

After the seminal work of Efron (1979), however, bootstrap methodologybecomes another sword for statistical inference because of its success in manycases. In fact, it is more efficient than the asymptotic methods in some situ-ations, since it converges faster to the exact distribution than the asymptoticapproximation does. The basic idea is quite similar in finding the exact dis-tribution by Monte Carlo simulations. Instead of generating a random samplefrom the distribution function F , the bootstrap method generates the randomsample from the empirical distribution function Fn which is a discrete proba-bility distribution that gives probability 1/n to each observed value xin

i=1. Inother words, a sample of size n from Fn is just equivalent to drawing a sample

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of size n with replacement from the collection xini=1. Now, let us call the

resampling methods depending on F and Fn as parametric and non-parametricbootstrap methods respectively. Since the parametric bootstrap depends on thedistribution function F , one could expect a good approximation of the exactdistribution for a given sample size as long as the replicates are large enough.For the non-parametric bootstrap, it highly depends on the observed valuesxin

i=1. If we have a "perfect" sample from F , then a good performance of thebootstrap method can be anticipated. While, it is almost impossible to evalu-ate whether a sample is "perfect" or not, we could pin our hope on the largesample size again. We could expect that the distribution of T ∗n converges to theasymptotic distribution of Tn when n is large enough, and this is called the va-lidity of the bootstrap method. Furthermore, under some conditions, we couldalso expect that the approximation of the exact distribution by the bootstrapdistribution is better than the asymptotic distribution, and this is called thesecond order property. Beside the second order property, another attractive-ness of the bootstrap happens when the asymptotic distribution is inapplicabledue to the nuisance parameters which are difficultly addressed. This idea isessential in Paper III and IV.

As we said before, different statistical inferences, such as confidence in-tervals and hypothesis tests, can be implemented once we get the knowledgeof the distribution of statistic. To illustrate the bootstrap test, suppose we areinterested in doing hypothesis tests when the unknown θ belongs to parameterset Θ and H0 : θ ∈ Θ0 against θ ∈ Θ1, and the test statistic is Tn. Then thebasic steps of the bootstrap test can be listed as follows:

1. Calculate the test statistic by the observed values xini=1, say Tn(x) =

Tn(x1, ...,xn);2. Generate the bootstrap sample, x∗i n

i=1, in the way which is mentionedabove;

3. Calculate the test statistic by the bootstrap samples x∗i ni=1, say T ∗n (x) =

Tn(x∗1, ...,x∗n);

4. Repeat step (2)-(3) B times and obtain the bootstrap replicates, T ∗n,bwhere b = 1,2, ...,B

5. Get the critical value from the bootstrap replicates, and do the hypothesistest. Or do the hypothesis test by estimating the P-value by

P = B−1B

∑b=1

IT ∗n,b(x)<Tn(x)

(1.15)

where I is the identity function.

15

1.7 Robust block bootstrap panel unit root testsAs discussed in Section 1.5, after the critique of the unrealistic assumptionof cross-sectional independence, many studies have shifted their interest tothe handling of cross-sectional dependence from the original purpose. Eventhough many models with complicated cross-sectional dependence have beenproposed, the existing models still have many restrictions which are difficultto check by statistical tests. For example, Bai and Ng (2004) considered therelatively general model, however, dependence between the common factorsand idiosyncratic errors is not allowed, and the independence between the id-iosyncratic errors must be restricted if one wants to do a pooled test on theidiosyncratic errors. Furthermore, the cross-sectional dependence which isintroduced by the dynamic dependence is seldom studied.

To consider a more general DGP with less restriction on the cross-sectionaldependence leads to a dilemma between the issue of over parameterizationsand the practicability of the test. For this dilemma, an expedient that appliesthe non-parametric block bootstrap algorithm provides a potential solution.Following this thinking, Palm, Smeekes and Urbain (2011) (PSU) generalizedthe residual based block bootstrap theory which is proposed by Paparoditisand Politis (2003) (PP) and designed for univariate time series into a panelsetting. In such a way, the block bootstrap algorithm based panel unit roottests can be applied to the DGPs which contain a wild range of cross-sectionaldependence, and many existing models which are applied in panel unit roottest can be treated as a special case of PSU. They assume that yt is generatedas

yt = ΛFt +wt (1.16)

where the factor loadings, Λ =(λ1, ...,λd)′, the common factors, Ft =(F1,t , ...,Fd,t)′

and the idiosyncratic error, wt = (w1,t , ...,wN,t)′. Let the factor and idiosyn-cratic components be generated by

Ft = ΦFt−1 + ftwt = Θwt−1 + vt

where Φ = diag(φ1, ...,φd) and Θ = diag(θ1, ...,θN) . vt and ft are generatedby (

vtft

)= ψ (L)εt =

[ψ11 (L) ψ12 (L)ψ21 (L) ψ22 (L)

][εv,tε f ,t

](1.17)

where the lag polynomial ψ (z) = ∑∞j=0 ψ jz j and ψ0 = I. The coefficients of

the lag polynomial satisfy some regular conditions. εt are i.i.d. with Eεt = 0,Eεtε

′t = Σ and E |εt |2+ε < ∞ for some ε > 0. This DGP is more general than

the existing models in many ways, for example, normally they assumed thatthe common factors and idiosyncratic errors are independent. However, it isnot the case in PSU, and more importantly, almost all studies assume that thematrix of the coefficients of the lag polynomial, ψ11, are diagonal so that the

16

cross-sectional dependence which is caused by dynamic dependence is wiped,however, this is also not the case in PSU.

Retaining the original purpose of panel unit root tests, PSU considers thesimple null hypothesis that each cross section unit contains a unit root, and thisnull hypothesis can be implied by different combinations of non-stationarityof common factors and idiosyncratic errors. To understand the main ideasbehind their tests, an important issue must be clarified. The following auxiliaryregression is introduced when they construct the test statistics.

∆yit = ρiyi,t−1 + eit (1.18)

Note that there are no ρi in the DGP, however, according to the null hypothesisand following the spirit of LLC and IPS, two test statistics can be constructedby the pooled OLS estimation of ρi and the average of OLS of ρi from eachi respectively. Say τp and τgm. Their asymptotic results indicate that two teststatistics converge to some limiting distributions for fixed N. Since too manynuisance parameters are present in the limiting distribution, there is no doubtthat it can not applied for statistical inference. While the non-parametric tool,the block bootstrap algorithm can be applied to mimic the exact distribution ofthose two statistics. By the theoretical justification of PSU, the block bootstrapbased panel unit root tests are asymptotically valid, and the simulation studiesindicate that their tests have robustness against a wide range of cross-sectionaldependence when comparing with LLC and IPS.

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2. Summary of the papers

2.1 Paper I: Measure of location based estimators insimple linear regression.

We consider certain measure of location-based estimators (MLBEs) for theslope parameter in a linear regression model with a single stochastic regressor.The median-unbiased MLBEs are interesting as they can be robust to heavy-tailed samples. As we have discussed, the classical LSE is very sensitive tolarge values of the error term. We also introduced the idea that the estimatorswhich are robust to heavy tail error distributions can be obtained using non-parametric or distribution free techniques. In this paper, we study the robustMLBEs for the slope parameter in a regression model and investigate theirfinite-sample and asymptotic properties in a parametric setting.

We consider any estimator β for β that can be decomposed into β = β +medz1,z2, . . . ,zk, where the zi are i.i.d. continuous random variables withzero median and k is odd. For an example of an MLBE, consider the incom-plete pairwise-slope estimator for β based on a sample of size n

βPS = medy2− y1

x2− x1,y4− y3

x4− x3, . . . ,

y2k− y2k−1

x2k− x2k−1

(2.1)

= β +medz1,z2, . . . ,zk,

wherezi =

u2i−u2i−1

x2i− x2i−1,

and medz1,z2, . . . ,zk is the sample median of z1,z2, . . . ,zk. If the zi are i.i.d.continuous random variables, standard results for order statistics show that theexact distribution of βPS−β when k is odd can be expressed in terms of theincomplete beta function

G(z;k) =[Fz(z)

]r+1r

∑s=0

(r + s

r

)[1−Fz(z)

]s=

Γ(k +1)Γ(r +1)Γ(r +1)

∫ Fz(z)

0tr(1− t)rdt, (2.2)

where Γ(·) is the gamma function, Fz(·) is the cdf of the zi and k = 2r + 1.Another example of a MLBE that we consider is

βUF = medy1−µy

x1−µx,y2−µy

x2−µx, . . . ,

yn−µy

xn−µx

, (2.3)

18

where µy and µx are location parameters of the yi and xi, respectively. We willsometimes refer to this estimator as unfeasible as it requires both µy and µx tobe known, which for most cases will not be realistic (cf. the b(α) estimatorsof Blattberg and Sargent, 1971). Then, in view of Equation (2.2), it is readilyshown that the median of β −β is zero also. Hence, β is a median-unbiasedestimator for β . If, in addition, the density of the zi is symmetric about zero,then so is that of β − β . This tells us that the distribution of β is centeredabout the unknown parameter β . Furthermore, if the median of zi is unique (ingeneral, the median may be an interval instead of a single number), then thesample median is a consistent estimator for the population median (e.g. Jiang,2010, p. 5) and β converges in probability to β as k tends to infinity.

About the heavy tail behavior, we apply a symmetric stable distributionor a normal mixture to describe it such that the problems can be parameter-ized. More specifically, we focus on the case where the explanatory variable,which is assumed to be stochastic, follows a symmetric stable distribution andthe error is either symmetric stable, with the same index of stability as theexplanatory variable, or a normal mixture. In addition, we also consider aconditionally heteroscedastic specification. Then for the different models, weestablish different conditions under which Equation (2.1) and Equation (2.3)are consistent, median-unbiased estimators with exact distribution that can beexpressed in terms of Equation (2.2), and with exact densities that are sym-metric about β .

2.2 Paper II: Panel unit root tests based on the samplevariance.

Since unit root tests have poor power properties especially for small samplesizes, many researchers consider borrowing the strength from cross sectionunits to improve the power performance. However, no matter the first genera-tion or the second generation panel unit root test, almost all of them are smallmodifications of Dickey-Fuller tests. Thus, in this paper, we propose a setof tests for the unit root in an entirely different way when the cross-sectionaldimension is considered.

Compared to the standard statistical methods, there is a special and trickysituation in time series econometrics. That is we can not observe the realiza-tion of one process several times repeatedly. It implies that we can not directlystudy the properties of expectation and variance of economic variables at anygiven time point. However this impossible issue becomes "feasible" in a panelsetting, since one economic variable can be repeatedly observed from differentregions or countries. Assuming the economic variable has similar propertiesamong all cross section units, then the properties of expectation and varianceat each time point can be applied. On the other hand, it is well known thatthe first-order autoregressive process is weakly stationary for a suitable choice

19

Figure 2.1. Indicating graph of sample variances at each time points of simulated panel data. Repeatedlygenerate simple AR(1) process with standard Gaussian innovations, then calculate the sample variance foreach fixed time points and draw the time series graphs for each cases. Left panel: The proper distributionof the initial values are given. Right panel: The initial values are fixed as 0

of the distribution of its initial value, provided that the autoregressive coeffi-cient is less than one in absolute value. Thus, the basic idea of our tests is thatthe trajectory of the cross sectional sample variance behaves differently forstationary than for non-stationary processes. Figure 2.1 gives an illustration.Given this idea, we construct the framework of our method as follows: Forsimplicity, consider the DGP

yit = ρiyit−1 + εit (2.4)

where i = 1, ...,N and t = 1, ...,T . Assume that εit are independent standardGaussian noise, i.e. εit ∼N (0,σ2), and the initial values are fixed as 0, i.e.yi0 = 0 for all i. Consider the null hypothesis

H0 : ρ1 = ... = ρN = 1 (2.5)

and the alternative hypothesis

H1a : ρ1 = ... = ρN ≡ ρ and |ρ|< 1 (2.6)

orH1b : |ρi|< 1 (2.7)

for all i. The cross sectional sample variance at time t is

S2t =

1N

N

∑i=1

(yit − yt)2 =

1N

N

∑i=1

y2it − y2

t . (2.8)

20

First, it is natural to consider the variation of the variance at each time point todistinguish the unit root process from the stationary process. The test statisticis

ψ =

√∑

Tt=1

(S2

t −S2)2

/T

S2(2.9)

Secondly, we consider fitting a straight line to go through each circle point inthe right panel of Figure 2.1. For the unit root case, those points can be fit bythe straight line very well, however, it is not so in the stationary case. Then weuse the R2 and F statistics to measure the goodness of fit, thereby they can beapplied to do hypothesis tests on unit roots. Consider the auxiliary model

S2t = β0 +β1t +ut (2.10)

and the test statistics

ψR2 =β 2

1 ∑Tt=1 (t− t)2

∑Tt=1

(S2

t −S2)2 (2.11)

and

ψF =β 2

1 ∑Tt=1 (t− t)2

∑Tt=1

(S2

t − S2t

)2/(T −2)

. (2.12)

The statistic ψR2 should be close to 1 under the null hypothesis, and close to0 for the stationary case. Similarly, under the null hypothesis, the statistic ψFshould be larger than in the case when the processes are stationary. Further-more, our test statistics all have an interesting property which can be illustratedby simulation, that is, the distribution of the test statistics is robust to a partic-ular covariance structure of the cross section units. The particular covariancestructure is the so called "equal correlation" i.e. the covariance matrix can beformulated as

Σ =

1 τ · · · τ

τ. . . . . .

......

. . . . . . τ

τ · · · τ 1

(2.13)

About the asymptotic results, we show that the sequential limiting distri-butions of ψ and ψR2 are normal. For the asymptotic distribution of ψF , wefind a non-standard limiting distribution which can be described in terms offunctionals of a Gaussian process.

21

2.3 Paper III: On the robustness of the block panel unitroot test

Although many panel unit root tests which consider the model with rathercomplicated cross-sectional dependence structure have been proposed in theliterature, the existing models still have many restrictions which are difficultto confirm by statistical tests or economic theory. In order to fill those gaps,PSU developed block bootstrap based panel unit root tests which can be ap-plied under a more general assumption of the cross-sectional dependence. Insuch a way, most of the existing models can be treated as special cases of PSU.

The asymptotic validity of block bootstrap panel unit root tests has been rig-orously proved in PSU. However, the small sample properties have not beenfully investigated. More specifically, they only compared their tests with twopanel unit root tests from the first generation, and the DGPs of their simu-lations do not fully display the generality as made by their assumptions, forexample, they only considered the case in which common factors and idiosyn-cratic errors are not dependent neither in the sense of wide range of plausibledynamic dependencies nor the contemporaneous dependence. Moreover, thereare some other detailed issues which are not concerned as well. For example,they did not consider the case in which the common factors are I(0) and id-iosyncratic errors are I(1) and the multi-factors case is also not included. Thusin this study, we do a further investigation on the small sample properties ofthe PSU by comparison with two other panel unit root tests from the secondgeneration and under a more general and complicated DGP. Specifically, thePSU test will be applied on the data which are generated by the model consid-ered in Bai and Ng (2004) and Chang (2004) to investigate the small sampleperformance under some relatively restrictive assumptions. Meanwhile, PSUtests and those two referred tests will also be applied to the data which is gen-erated by the PSU’s DGP with more general cross-sectional dependence toexactly see the robustness of PSU. Those issues are also studied in this paper.Based on our simulation results, we have the following main conclusions:

1. Under the specific DGPs of Chang (2004) and Bai and Ng (2004), thePSU test, especially τgm has generally as good size and power propertiesas Chang and BN tests, except in the case when negative moving aver-age coefficients are present. In the case with negative moving averagecoefficients, both τp and τgm have extreme size distortions.

2. Under the DGP of PSU with general cross-sectional dependence struc-ture, the PSU test exhibits robustness with good size and power proper-ties when the idiosyncratic error and common factor part have the sameintegrated order, and better than the other two methods.

3. About the case in which the common factor and idiosyncratic error havea different integrated order, the PSU test is oversized. This problem ismore severe in the case where common factors contain unit roots.

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4. In general, the group mean test τgm is more robust than the pooled testτp. In both the DGP of Chang (2004) and Bai and Ng (2004), when theARMA coefficients are randomly chosen from U [−0.8,0.8], τp exhibitssevere size distortion.

2.4 Paper IV: Block bootstrap panel unit root tests withdeterministic terms

Based on the block bootstrap method, PSU proposed panel unit root testswhich are robust against a wide range of cross-sectional dependence. How-ever several open problems are left and one of the most important issues isthe handling of deterministic terms, especially for empirical studies. Thus, weaim to generalize the block bootstrap panel unit root tests in the sense of con-sidering deterministic terms in the model in this paper. Let xt = (x1,t , ...,xN,t)for t = 1, ...,T be generated by

xt = dmt + yt , (2.14)

where dmt =

(dm

1,t , ...,dmN,t

)′for m = 0,1,2 is the deterministic part. For each

i, dmi,t = β m′

i zmt , where

zmt =

01

(1, t)′

if m = 0if m = µ

if m = τ

and βmi =

0

β1i(β1i,β2i)

′

if m = 0if m = µ

if m = τ.(2.15)

Then dmt can be expressed as dm

t = β mzmt where β m = (β m

1 , ...,β mN )′. For future

reference, we also partition β τ in another way and define β τ = (β1·,β2·) whereβ j· =

(β j1, ...,β jN

)′ and j = 1,2. yt is the stochastic part and follows the samemodel as in PSU, see Section 1.7.

We propose two different strategies to deal with the deterministic terms.One is to modify the test statistics by adding the deterministic terms into theauxiliary Dickey-Fuller regression and adjust the bootstrap algorithm regard-ing the model specification. More specifically, instead of applying the auxil-iary regression (1.18), we consider the following auxiliary regressions giventhe model specification. For model µ ,

∆xit = α +ρixi,t−1 + eit . (2.16)

For model τ ,∆xit = α +β t +ρixi,t−1 + eit . (2.17)

Then the test statistics are constructed by the corresponding pooled OLS esti-mation of ρi and average of the OLS estimation of ρi from each i. Besides the

23

modification of the test statistics, we modify the bootstrap algorithm by gener-alizing the residual based block bootstrap theory proposed by PP. The secondstrategy is applying a certain detrending method on the observations and thenperforming the block bootstrap panel unit root tests on the detrended residuals,i.e. a two stage method. More specifically, we apply the same test statistics butmodify the bootstrap algorithm based on the suggestions which are proposedby Smeekes (2009). The modifications of the bootstrap algorithm mainly con-cern the detrending step, and can be summarized as the following: (i) the datamust be detrended not only for the construction of the independent bootstrapresiduals but also for calculating the test statistics based on bootstrap samples;(ii) the detrending procedures for calculating the test statistics based on theoriginal sample and bootstrap sample must be identical; (iii) the detrendingprocedure for constructing the bootstrap residuals can be different from an-other two, however some conditions of the convergence rate must be satisfied.About the selection of detrending methods, full sample OLS detrending, GLSdetrending and recursive detrending are considered in this paper, however, weonly provide justification of asymptotic consistency for OLS detrending underthe main null hypothesis.

The simulation results show that all tests have acceptable size properties.The tests which are based on detrending methods have better power than thetests which are based on the conventional methods. Furthermore the testsbased on the GLS detrending method have the best power among all detrend-ing methods. About the difference between pooled and group mean statistics,even though group mean statistics have better power than pooled in general,we still recommend pooled statistics because of the unexpected results of sizeof group mean statistics in some cases. At last, we provide an applicationwhich illustrates the use of the tests.

24

3. Further research

About the first paper, we only create the results for the simple linear regres-sion. Besides generalizing it into the multivariate case, other potential furtherresearch is to apply those results to the unit root test. The main problems couldbe concentrated on how to deal with the dependency structure.

About the second paper, we only considered the simplest case. Next, thereare several substantial directions to generalize this idea. First, we will considerincluding deterministic terms which are not identical for all cross section units.So far, we only consider the simple AR(1) process without drift term for eachcross section unit. Despite the lack of realism in an empirical study, it isstill worth mentioning that the properties of cross section sample variances ateach time point will not be affected by the drift terms if we assume that thedrift terms are identical through all cross section units. More generally, weconsider the model as

yit = Dt +ρyi,t−1 + εit (3.1)

where Dt denotes the deterministic terms. Thus the cross section sample vari-ance at each time is invariant with Dt , no matter how complicated it is. How-ever, if we consider AR(1) processes with non-identical deterministic terms,then the distribution of the test statistic will depend on the variation of the co-efficients of deterministic terms. Therefore, we will consider an AR(1) processwith non-identical drift terms first.

Second, we will consider more general stochastic processes, for examplewith non-normal innovations and serial correlations. Third, even though ourmethods have robustness to a particular covariance structure, they still shouldbe called first generation panel unit root tests. Thus how to deal with cross sec-tion dependence in this framework could be an interesting problem. For thispoint, we try to handle the cross-sectional dependence under the single factorassumption in other research. Furthermore, in most cases, we only create theasymptotic results in a sense of sequential limits. It is still worth finding thejoint limiting distribution.

Regarding the third paper, there are two further problems which are worthstudying. First, the study indicates that all tests have more or less size distor-tion when the coefficient of moving average is negative for all cross-sectionunits. This problem is well studied in the univariate unit root test, for example,Ng and Perron (2001), however, there is no study on this problem in a panelsetting. So far, it is still difficult to give a proper solution, but we can see thatchoosing the number of lag terms by some better criteria is the most important

25

step. Second, for the pooled tests which are based on PANIC residuals, theassumption of independence of idiosyncratic errors is always necessary. How-ever, we could relax this assumption by applying some other test which is freeof the dependence, e.g. the robust t test by Breitung and Das (2005), to thePANIC residuals. We proceed with this idea in future research.

26

4. Acknowledgements

Many people have in different ways contributed to this work. I would like to takethis special opportunity to express my sincere gratitude to all those who gave me thepossibility to complete this thesis.

I would like to express my sincere gratitude to my supervisor, Prof. Rolf Larssonwithout whom the thesis would not have been completed. I appreciate all of yourgenuine supports on my studies. Your conscientious attitude to the research will al-ways encourage me in my whole academic career. This is a "marathon", and I willendeavour to finish it with your suggestions and encouragement.

I wish to send my appreciation to my assistant supervisor, Prof. Johan Lyhagen.I was always enlightened after the discussion with you, and you always cheered meup with enthusiasm and positive attitude. I want to thank Prof. Fan Yang Wallentin,who has given me the precious opportunity to start my academic life, for being alwaysavailable for advice. Also, I would like to thank Daniel Preve who supervised and co-operated with me during the PhD studies; Prof. Sune Karlsson, who was the opponentof my licentiate thesis, for your heuristic discussion; Prof. Soren Johansen and Prof.Niels Haldrup for your kindness help and providing me so many useful suggestionsduring my visit to CREATES at Aarhus University. Prof. Changli He and AssociateProf. Martin Sköld, for your kindness help and guidance during my Master degreestudies.

In particular, my thanks also go to all of my colleagues from the department. Lis-beth Hansson, thank you for your patient help about my teaching jobs; Lars Forsberg,thank you for your sincere care all the time; Inger Persson, your warm smiles alwayscheered me up especially in the dark winter; Bo Wallentin, you are always gentle to ev-eryone; Tommy Perlinger, do you want to play Chinese chess with me; Ahmad, Rauf,thank you for your nice suggestion to my thesis; Katrin Kraus, thank you for yourkindness help and discussions; Davoud Emamjomeh and Lars-Göran Svensk, thankyou for your technical support; Eva Enefjord and Eva Karlsson, finally I can stopdisturbing you. Wait, maybe not...:) I also want to say thank you to all of the PhDcandidates in our department. All of you guys are outstanding and I learned so manyfrom you.

I would like to say tusen tack to all of my friends in Sweden. Prof. ElisabethSvensson, I think Öland is the most beautiful island all over the world; Roland, Ibelieve your Chinese must be much better than my Swedish; Birgitta and Sven, wehope to get more sweet tomatoes from your lovely garden in this summer; AndersW-Löfgren, I never forget the medicine for seasickness that you gave me on the boatfrom Stockholm to Helsinki in 2007.

I also want to say thank you to all of my classmates since the Master program.These include Haishan Yu, Jianxin Wei, Shaobo Jin and Xingwu Zhou at Uppsala

27

University; Xia Shen and Ying Li at Swedish University of Agricultural Sciences;Chengcheng Hao, Ying Pang and Yuli Liang at Stockholm University; Deliang Daiat Linnaeus University; Shutong Ding and Yishen Yang at Örebro University; Dao Liat Dalarna University; Qi Cao at Groningen University; Feng Li at Central Universityof Finance and Economics; Hao Luo at Tsinghua University. Your guys made me awonderful life in Sweden.

Di, Lei, Long, Meng, Qi, Wei, Xuan my diehard followers since the college timeand all of your families, I wish I could put all of you in my pocket and bring youwherever I go, so that I can share my happiness with you guys all the time. Let’s 2together endlessly.

My unbounded thanks go to my family. My parents, you gave me life, brought meup and provided me whatever I wanted. You let me know how to be a good person inthis amazing world. Siyi, my son, changing your dippers always helped me to switchmy brain when I was stuck on the tricky problems. Finally, I am grateful to my wife,Xin Zhao. It is your patient, trust and love support me to approach the success in myPhD studies. I think it is the perfect place to write down: I love you!

Xijia LiuApril 8, 2014

Carolina Rediviva, Uppsala.

28

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A doctoral dissertation from the Faculty of Social Sciences,Uppsala University, is usually a summary of a number ofpapers. A few copies of the complete dissertation are keptat major Swedish research libraries, while the summaryalone is distributed internationally through the series DigitalComprehensive Summaries of Uppsala Dissertations from theFaculty of Social Sciences. (Prior to January, 2005, the serieswas published under the title “Comprehensive Summaries ofUppsala Dissertations from the Faculty of Social Sciences”.)

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